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Matteo Capucci (he/him) (Jan 21 2021 at 08:28):Hi people, I opened a question on StackExchange but maybe here I'll have more luck.
Is the inverse image of a separated presheaf separated? It seems so. My proposed proof is linked above.
Morgan Rogers (he/him) (Jan 21 2021 at 10:28):working on it
Jens Hemelaer (Jan 21 2021 at 10:34):@Matteo Capucci (he/him) I just looked at it and your proof works, as far as I can see. I added a comment on StackExchange.
@Morgan Rogers (he/him), maybe you can spot something that I missed?
Morgan Rogers (he/him) (Jan 21 2021 at 10:46):I'm writing up a higher-tech proof :wink:
Matteo Capucci (he/him) (Jan 21 2021 at 10:53):Thank you guys!! Though SE went down for maintenance for me :laughing: so I have to wait to check your answers
Matteo Capucci (he/him) (Jan 21 2021 at 10:54):I'm weirded out by the fact I couldn't find that fact on any book. Seems like a standard thing to notice.
Morgan Rogers (he/him) (Jan 21 2021 at 10:56):I reckon that's mostly due to a disinterest in separated presheaves in general, but now that you've said that I'll skim the relevant part of the Elephant
Morgan Rogers (he/him) (Jan 21 2021 at 11:04):Matteo Capucci (he/him) said:
Thank you guys!! Though SE went down for maintenance for me :laughing: so I have to wait to check your answers
Don't worry, it's taking a little bit because of some diagrams I need to typeset
Matteo Capucci (he/him) (Jan 21 2021 at 11:16):Morgan Rogers (he/him) said:
I reckon that's mostly due to a disinterest in separated presheaves in general, but now that you've said that I'll skim the relevant part of the Elephant
Oh well, I should learn to consult the Elephant before bothering people
Morgan Rogers (he/him) (Jan 21 2021 at 11:20):First mention is Example A2.6.3(d), but there is some further stuff on separated presheaves in there (I don't know about this result off the top of my head)
Jens Hemelaer (Jan 21 2021 at 11:31):Apparently, the category of separated presheaves is a quasitopos, so maybe the proof appears in the quasitopos literature.